Discrete state coupled to a continuum

[Heimisson]

Homework Statement

This is not so much a homework problem but a part of a project I'm working on.

So in just a few words; what I have (at time t=0) is a discrete state (half simple harmonic oscillator) connected to a wire with continuous states. These states are coupled by a complex coupling. My problem is that somehow I will need to normalize the wavefunction with the continuous states but in principle it can't be normalized.

Homework Equations

The Attempt at a Solution

I've tried to add to the wavefunction a factor $$e^{\epsilon x}$$ to make the integral converge at - infinity, because one could argue that if epsilon is small it shouldn't change the coupling and the coupling shouldn't be present very far away. But I'm not sure if this is the right method and it doesn't seem to give me anything good.

 

[Heimisson]

I actually just realized how this might work, but please leave an reply if you have an better idea or I'm wrong.